How to UV map complicated bevels
Unwrap final geometry, or split faces, or mark additional seams

You are unwrapped a model, which means to assign to evry face (5 in your case) a portion of the UV space.

By beveling the edges, your geometry doesn't have still 5 faces, but a bit more. The UV informations about those new faces are coming from an interpolations of the previous faces UV, so it may happen that not all of them are corrects (expecially when your UV is made by disjoined pieces) as in you case:



In addition your geometry has some caharecteristics that make the effect come out. If it has been more compact on the X axis you would not have such distortion.



Solutions:

1) Apply all modifers ( or converto to mesh your mesh... with Alt+C, or at least the ones that have more impact on the geometry, and THAN unwrap your geometry (Suggested if suitable)

2) If you need to keep the modifiers in your workflow you can try to disconnect the triangular faces. With vertex disjoined there will surely be no wrong interpolation between the triangular faces and the others (but you have to add additional vertex bevel modifiers to smooth the spikes..)

3) Mark all edge as seams. In this way Blender will consider each face separately and it will not occour unexpected interpolation and your topology will remain clean. Anyway your UVs islands may result slightely overlapped or disjoined, so you'll have to re-unwrap in anycase, but at least you can "preview" the UV correctly without applying modifiers.